Differential Galois Approach to the Non-integrability of the Heavy Top Problem

摘要

We study integrability of the Euler-Poisson equations describing the motionof a rigid body with one fixed point in a constant gravity field. Using theMorales-Ramis theory and tools of differential algebra we prove that asymmetric heavy top is integrable only in the classical cases of Euler,Lagrange, and Kovalevskaya and is partially integrable only in theGoryachev-Chaplygin case. Our proof is alternative to that given by Ziglin({\em Funktsional. Anal. i Prilozhen.}, 17(1):8--23, 1983; {\em Funktsional.Anal. i Prilozhen.}, 31(1):3--11, 95, 1997).